Strengthen the second q-supercongruence modulo [m]Φ_m(q)^2

Prove that the q-supercongruence in Theorem 2, namely the congruence for the parameterized sum involving [8n−1], q-shifted factorials with base q^4, and parameters m, t∈{1,3}, and s≤(tm+1)/4 satisfying m≡−t (mod 4), holds modulo [m]Φ_m(q)^2.

Background

The paper proves a one-parameter q-supercongruence corresponding to Van Hamme-type (F.2) congruences modulo Φ_m(q)^3. The authors conjecture a stronger modulus, again motivated by numerical evidence: the modulus should be the product of the q-integer [m] and Φ_m(q)^2.

Upon setting m=p^r and taking the limit q→1, the conjectured q-congruence would imply a corresponding p-adic extension of the hypergeometric congruence associated with (F.2), for prime powers satisfying p^r≡−t (mod 4), with t∈{1,3}.